Estimator properties

We estimate the unknown mean θ of a random variable X (where X has a finite and positive variance) by forming the sample mean Mn=(X1+⋯+Xn)/n of n i.i.d. samples Xi and then forming the estimator

Θˆ=Mn+1n.

Is this estimator unbiased?
unanswered

Is this estimator consistent?
unanswered

Consider now a different estimator, Θˆn=X1, which ignores all but the first measurement.

Is this estimator unbiased?
unanswered

Is this estimator consistent?
unanswered

1 answer

1 - No
2 - Yes
3 - Yes
4 - No